Where the known results stand

The reduction in plain words

When the first guess x = (p+3)/4 works

Full table for all 1181 primes

Checked against the table

What decides the second guess?

Our paper on distinct denominators

Half an answer

The gap isn't at most 7

Thanks, that settles it

[Demo] Our paper reduces Erdős–Straus to primes 

 and checks them below 100,000. Two things we couldn't explain: why the first guess 

 works for only about half of these primes, and whether the gap between the smallest working 

 stays small. Anything that explains either would help, including pointers to what's already known.

[Demo] Covering Erdős–Straus residue classes

[Demo] Our paper reduces Erdős–Straus to primes p \equiv 1 \pmod{24} and checks them below 100,000. Two things we couldn't explain: why the first guess x = (p+3)/4 works for only about half of these primes, and whether the gap between the smallest working x and (p+3)/4 stays small. Anything that explains either would help, including pointers to what's already known.